A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

GEMINUS 227
Attempt to prove the Parallel-Postulate.
Geminus devoted much attention to the distinction between
postulates and axioms, giving the views of earlier philosophers
and mathematicians (Aristotle, Archimedes, Euclid,
Apollonius, the Stoics) on the subject as well as his own. It
was important in view of the attacks of the Epicureans and
Sceptics on mathematics, for (as Geminus says) it is as futile
to attempt to prove the indemonstrable (as Apollonius did
when he tried to prove the axioms) as it is incorrect to assume
what really requires proof, as Euclid did in the fourth postu-
'
late [that all right angles are equal] and in the fifth postulate
[the parallel-postulate] '}
The fifth postulate was the special stumbling-block.
Geminus observed that the converse is actually proved by
Euclid in I. 17; also that it is conclusively proved that an
angle equal to a right angle is not necessarily itself a right
angle (e.g. the angle between the circumferences of two semi-
' '
circles on two equal straight lines with a common extremity
and at right angles to one another) ; we cannot therefore admit
that the converses are incapable of demonstration. 2
And
we have learned from the very pioneers of this science not to
'
have regard to mere plausible imaginings when it is a question
of the reasonings to be included in our geometrical
doctrine. As Aristotle says, it is as justifiable to ask scientific
proofs from a rhetorician as to accept mere plausibilities
from a geometer ... So in this case (that of the parallelpostulate)
the fact that, when the right angles are lessened, the
straight lines converge is true and necessary ; but the statement
that, since they converge more and more as they are
produced, they will sometime meet is plausible but not necessary,
in the absence of some argument showing that this is
true in the case of straight lines. For the fact that some lines
exist which approach indefinitely but yet remain non-secant
(dcrvfi7rTCQToi), although it seems improbable and paradoxical,
is nevertheless true and fully ascertained with reference to
other species of lines [the hyperbola and its asymptote and
the conchoid and its asymptote, as Geminus says elsewhere].
May not then the same thing be possible in the case of
1
Proclus on Eucl. I, pp. 178-82. 4; 183. 14-184. 10.
2 lb., pp. 183. 26-184. 5.
Q 2
228 SUCCESSORS OF THE GREAT GEOMETERS
straight lines which happens in the case of the lines referred
to? Indeed, until the statement in the postulate is clinched
by proof, the facts shown in the case of the other lines may
direct our imagination the opposite way. And, though the
controversial arguments against the meeting of the straight
lines should contain much that is surprising, is there not all
the more reason why we should expel from our body of
doctrine this merely plausible and unreasoned (hypothesis) ?
It is clear from this that we must seek a proof of the present
theorem, and that it is alien to the special character of
postulates.' l
Much of this might have been written by a modern
geometer. Geminus's attempted remedy was to substitute
a definition of parallels like that of Posidonius, based on the
notion of eqvAdistance. An-Nairizi gives the definition as
follows :
'
Parallel straight lines are straight lines situated in
the same plane and such that the distance between them, if
they are produced without limit in both directions at the same
time, is everywhere the same ', to which Geminus adds the
statement that the said distance is
the shortest straight line
that can be drawn between them. Starting from this,
Geminus proved to his own satisfaction the propositions of
Euclid regarding parallels and finally the parallel-postulate.
He first gave the propositions (1) that the 'distance ' between
the two lines as defined is perpendicular to both, and (2) that,
if a straight line is perpendicular to each of two straight lines
and meets both, the two straight lines are parallel, and the
'
distance ' is the intercept on the perpendicular (proved by
reductio ad absurdum). Next come (3) Euclid's propositions
I. 27, 28 that, if two lines are parallel, the alternate angles
made by any transversal are equal, &c. (easily proved by
drawing the two equal ' distances ' through the points of
intersection with the transversal), and (4) Eucl. I. 29, the converse
of I. 28, which is proved lyy reductio ad absurdum, by
means of (2) and (3). Geminus still needs Eucl. I. 30, 31
(about parallels) and I. 33, 34 (the first two propositions
relating to parallelograms) for his final proof of the postulate,
which is to the following effect.
Let A B, CD be two straight lines met by the straight line
1
Proclus on Eucl. I, pp. 192. 5-193. 3.